Theorem isphg 30836

Description: The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isphg.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
isphg	⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem isphg

Dummy variables 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ph 30832	. . 3 ⊢ CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))})
2	1	elin2 4203	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}))
3		rneq 5947	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
4		isphg.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
5	3, 4	eqtr4di 2795	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
6		oveq 7437	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
7	6	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔𝑦)) = (𝑛‘(𝑥𝐺𝑦)))
8	7	oveq1d 7446	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔𝑦))↑2) = ((𝑛‘(𝑥𝐺𝑦))↑2))
9		oveq 7437	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(-1𝑠𝑦)) = (𝑥𝐺(-1𝑠𝑦)))
10	9	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑠𝑦))))
11	10	oveq1d 7446	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2))
12	8, 11	oveq12d 7449	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)))
13	12	eqeq1d 2739	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
14	5, 13	raleqbidv 3346	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
15	5, 14	raleqbidv 3346	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
16		oveq 7437	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (-1𝑠𝑦) = (-1𝑆𝑦))
17	16	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑥𝐺(-1𝑠𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
18	17	fveq2d 6910	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑛‘(𝑥𝐺(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑆𝑦))))
19	18	oveq1d 7446	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2))
20	19	oveq2d 7447	. . . . . 6 ⊢ (𝑠 = 𝑆 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
21	20	eqeq1d 2739	. . . . 5 ⊢ (𝑠 = 𝑆 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
22	21	2ralbidv 3221	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
23		fveq1 6905	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺𝑦)) = (𝑁‘(𝑥𝐺𝑦)))
24	23	oveq1d 7446	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝑥𝐺𝑦))↑2))
25		fveq1 6905	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
26	25	oveq1d 7446	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2))
27	24, 26	oveq12d 7449	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
28		fveq1 6905	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑥) = (𝑁‘𝑥))
29	28	oveq1d 7446	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥)↑2) = ((𝑁‘𝑥)↑2))
30		fveq1 6905	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑦) = (𝑁‘𝑦))
31	30	oveq1d 7446	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑦)↑2) = ((𝑁‘𝑦)↑2))
32	29, 31	oveq12d 7449	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)) = (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))
33	32	oveq2d 7447	. . . . . 6 ⊢ (𝑛 = 𝑁 → (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
34	27, 33	eqeq12d 2753	. . . . 5 ⊢ (𝑛 = 𝑁 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
35	34	2ralbidv 3221	. . . 4 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
36	15, 22, 35	eloprabg 7543	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
37	36	anbi2d 630	. 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → ((⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
38	2, 37	bitrid 283	1 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⟨cop 4632  ran crn 5686  ‘cfv 6561  (class class class)co 7431  {coprab 7432  1c1 11156   + caddc 11158   · cmul 11160  -cneg 11493  2c2 12321  ↑cexp 14102  NrmCVeccnv 30603  CPreHil_OLDccphlo 30831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-ph 30832

This theorem is referenced by:  cncph  30838  isph  30841  phpar  30843  hhph  31197